Calculus - diagnostic tests

Success in calculus depends to a large extent on knowledge of the mathematics that precedes calculus: algebra, analytic geometry, functions, and trigonometry. The fol- lowing tests are intended to diagnose weaknesses that you might have in these areas.

After taking each test you can check your answers against the given answers and, if necessary, refresh your skills by referring to the review materials that are provided.

1. Evaluate each expression without using a calculator.

(a) (b) (c)

(d) (e) (f)

2. Simplify each expression. Write your answer without negative exponents.

(a) (b) (c)

3. Expand and simplfy.

(a) (b)

(c) (d)

(e)

4. Factor each expression.

(a) (b)

(c) (d)

(e) (f)

5. Simplify the rational expression.

(a) (b)

(c) (d)

y x x

y 1 y 1

x x²

x²4 x1 x2

2x²x1

x²9 ⴢ x3 2x1 x²3x2

x²x2

x³y4xy 3x^3兾29x^1兾26x^1兾2

x⁴27x x³3x²4x12

2x²5x12 4x²25

共x2兲³

共2x3兲²

(

^sa sb

)(

^sa sb

)

共x3兲共4x5兲 3共x6兲4共2x5兲

冉

^{3xx2y3兾21y兾32}

冊

²

共3a³b³兲共4ab²兲² s200 s32

16^3兾4

冉

²³

冊

²

5²³ 5²¹

3⁴ 3⁴

共3兲⁴ D I A G N O S T I C T E S T : A L G E B R A

A

xxiv

6. Rationalize the expression and simplify.

(a) (b)

7. Rewrite by completing the square.

(a) (b)

8. Solve the equation. (Find only the real solutions.)

(a) (b)

(c) (d)

(e) (f)

(g)

9. Solve each inequality. Write your answer using interval notation.

(a) (b)

(c) (d)

(e)

10. State whether each equation is true or false.

(a) (b)

(c) (d)

(e) (f)

^1兾x

a兾xb兾x 苷 1 ab 1

xy 苷 1 x 1

y

1TC

C 苷1T

sa²b² 苷ab

sab 苷sasb 共pq兲²苷p²q²

2x3 x1 1

ⱍ

^x4

ⱍ

³

x共x1兲共x2兲0

x²2x8 453x17

2x共4x兲^1兾23s4x 苷0

3

ⱍ

^x4

ⱍ

^苷10

x⁴3x²2苷0

2x²4x1苷0 x²x12苷0

2x

x1 苷 2x1 x x5苷14¹2x

2x²12x11 x²x1

s4h 2 h s10

s5 2

6. (a) (b)

7. (a) (b)

8. (a) (b) (c)

(d) (e) (f)

(g)

9. (a) (b)

(c) (d)

(e)

10. (a) False (b) True (c) False

(d) False (e) False (f) True

共1, 4兴 共1, 7兲

共2, 0兲傼共1, 兲 共2, 4兲

关4, 3兲

12 5

2 3, ²²3

1, s2 1¹²s2

3, 4 1

6

2共x3兲²7

(

x¹²

)

²³4

1 s4h 2 5s22s10

1. (a) (b) (c)

(d) (e) (f)

2. (a) (b) (c)

3. (a) (b)

(c) (d)

(e)

4. (a) (b)

(c) (d)

(e) (f)

5. (a) (b)

(c) 1 (d) 共xy兲

x2

x1 x3 x2

x2

xy共x2兲共x2兲 3x^1兾2共x1兲共x2兲

x共x3兲共x²3x9兲 共x3兲共x2兲共x2兲 共2x3兲共x4兲 共2x5兲共2x5兲

x³6x²12x8

4x²12x9 ab

4x²7x15 11x2

x 9y⁷ 48a⁵b⁷

6s2

1 8 9

25 4

1

81 81

81

If you have had difficulty with these problems, you may wish to consult the Review of Algebra on the website www.stewartcalculus.com.

A N S W E R S TO D I AG N O S T I C T E S T A : A L G E B R A

1. Find an equation for the line that passes through the point and (a) has slope

(b) is parallel to the -axis (c) is parallel to the -axis (d) is parallel to the line

2. Find an equation for the circle that has center and passes through the point . 3. Find the center and radius of the circle with equation . 4. Let and be points in the plane.

(a) Find the slope of the line that contains and .

(b) Find an equation of the line that passes through and . What are the intercepts?

(c) Find the midpoint of the segment . (d) Find the length of the segment .

(e) Find an equation of the perpendicular bisector of . (f) Find an equation of the circle for which is a diameter.

5. Sketch the region in the -plane defined by the equation or inequalities.

(a) (b)

(c) (d)

(e) x²y²4 (f) 9x²16y²苷144

y x²1 y1¹2x

ⱍ

^x

ⱍ

^{4 and}

ⱍ

^y

ⱍ

²

1y3

xy

AB AB AB

AB

B A B A B共5, 12兲

A共7, 4兲

x²y²6x10y9苷0 共3, 2兲 共1, 4兲

2x4y苷3 y

x 3

共2, 5兲 D I A G N O S T I C T E S T : A N A LY T I C G E O M E T RY

B

5.

y

x

1 2

0

y

0 x

y

0 4 x

3

_1

2 y

x 0

y

0 4x

_4

y

0 2 x 1

(a) (b) (c)

(d) (e) (f )

_1 3

2

_2

y=≈-1

≈+¥=4

y=1- x¹₂

1. (a) (b)

(c) (d)

2.

3. Center , radius 5 4. (a)

(b) ; -intercept , -intercept

(c) (d) (e)

(f)共x1兲²共y4兲²苷100 3x4y苷13

20

共1, 4兲 x 4 y ¹⁶3

4x3y16苷0 3⁴

共3, 5兲

共x1兲²共y4兲²苷52

y苷¹2x6 x苷2

y苷5 y苷3x1

A N S W E R S TO D I AG N O S T I C T E S T B : A N A LY T I C G E O M E T RY

If you have had difficulty with these problems, you may wish to consult the Review of Analytic Geometry on the website www.stewartcalculus.com.

1. The graph of a function is given at the left.

(a) State the value of . (b) Estimate the value of . (c) For what values of is ?

(d) Estimate the values of such that . (e) State the domain and range of .

2. If , evaluate the difference quotient and simplify your answer.

3. Find the domain of the function.

(a) (b) (c)

4. How are graphs of the functions obtained from the graph of ?

(a) (b) (c)

5. Without using a calculator, make a rough sketch of the graph.

(a) (b) (c)

(d) (e) (f)

(g) (h)

6. Let

(a) Evaluate and . (b) Sketch the graph of .

7. If and , find each of the following functions.

(a) fⴰt (b) tⴰf (c) tⴰtⴰt

t共x兲苷2x3 f共x兲苷x²2x1

f f共1兲

f共2兲

f共x兲苷

再

^{12xx21 ifif xx00 y苷1x}

1

y苷2^x

y苷2sx y苷sx

y苷4x²

y苷共x2兲³3 y苷共x1兲³

y苷x³

y苷f共x3兲2 y苷2f共x兲1

y苷f共x兲

f

h共x兲苷s4x sx²1 t共x兲苷 s³x

x²1 f共x兲苷 2x1

x²x2

f共2h兲f共2兲 f共x兲苷x³ h

f

f共x兲苷0 x

f共x兲苷2 x

f共2兲 f共1兲

f D I A G N O S T I C T E S T : F U N C T I O N S

C

6.(a) 7. (a)

(b) (b)

(c) 共tⴰtⴰt兲共x兲苷8x21 共tⴰf兲共x兲苷2x²4x5

y

x _1 0

1

共fⴰt兲共x兲苷4x²8x2 3, 3

(h) y

x 0 1

1

(g) y

x 0 _1 1

(f ) y

x 0 1

(e) y

x 0 1

(d) y

x 0 4

2

1. (a) (b) 2.8

(c) (d)

(e) 2.

3. (a) (b) (c)

4. (a) Reflect about the -axis

(b) Stretch vertically by a factor of 2, then shift 1 unit downward (c) Shift 3 units to the right and 2 units upward

5. (c) y

x 0

(2, 3) y

x 0

(a) (b) y

1

1 0 x

1 _1

x 共, 1兴傼关1, 4兴

共, 兲

共, 2兲傼共2, 1兲傼共1, 兲 126hh²

关3, 3兴, 关2, 3兴

2.5, 0.3 3, 1

2 y

0 x

1 1

FIGURE FOR PROBLEM 1

A N S W E R S TO D I AG N O S T I C T E S T C : F U N C T I O N S

If you have had difficulty with these problems, you should look at Sections 1.1–1.3 of this book.

1. Convert from degrees to radians.

(a) (b)

2. Convert from radians to degrees.

(a) (b)

3. Find the length of an arc of a circle with radius 12 cm if the arc subtends a central angle of . 4. Find the exact values.

(a) (b) (c)

5. Express the lengths and in the figure in terms of .

6. If and , where and lie between and , evaluate .

7. Prove the identities.

(a) (b)

8. Find all values of such that and .

9. Sketch the graph of the function y苷1sin 2xwithout using a calculator.

0x2 sin 2x苷sin x

x 2 tan x

1tan²x 苷sin 2x tan sin cos 苷sec

sin共xy兲 2

0 y

x sec y苷⁵4

sin x苷¹3

b

a

sec共5兾3兲 sin共7兾6兲

tan共兾3兲

30 2

5兾6

18 300

6.

8.

9.

_π 0 π x

2 y

0, 兾3, , 5兾3, 2

1

15

(

46s2

)

1. (a) (b)

2. (a) (b)

3.

4. (a) (b) (c)

5. (a)24 sin (b) 24 cos ¹2 2 s3

2 cm

360兾⬇114.6 150

兾10 5兾3

D I A G N O S T I C T E S T : T R I G O N O M E T RY

D

a

¨ b 24

F I G U R E F O R P R O B L E M 5

A N S W E R S TO D I AG N O S T I C T E S T D : T R I G O N O M E T RY

If you have had difficulty with these problems, you should look at Appendix D of this book.